## The binomial option pricing model – part 1

This is post #1 on the binomial option pricing model. Even though this is post #1, there are two previous posts with examples to illustrate how to price options using the one-period binomial pricing model (example of call and example of put). The purpose of post #1:

Post #1: Describe the option pricing formulas in the one-period binomial model.

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The one-period binomial option pricing model

We first consider the pricing of options on stock. The most important characteristic of the binomial option pricing model is that over a period of time, the stock price is assumed to follow a binomial distribution, i.e. the price of the stock can only take on one of two values – an upped value and a downed value. In this post, we describe how to price an option on a stock using this simplifying assumption of stock price movement.

Consider a stock with the following characteristics:

• The current share price is $S$.
• If the stock pays dividends, we assume the dividends are paid at an annual continuous rate at $\delta$.
• At the end of a period of length $h$ (in years), the share price is either $S_h=uS$ or $S_h=dS$, where $u$ is the up factor and $d$ is the down factor. The factor $u$ can be interpreted as one plus the rate of capital gain on the stock if the stock goes up. The factor $d$ can be interpreted as one plus the rate of capital loss if the stock goes down.
• If $\delta>0$, the end of period share price is $S_h=uS e^{\delta h}$ or $S_h=dS e^{\delta h}$. This is to reflect the gains from reinvesting the dividends. Of course if $\delta=0$, the share prices revert back to the previous bullet point.

The end of period stock prices are shown in the following diagram, which is called a binomial tree since it depicts the 2-state stock price at the end of the option period.

$\text{ }$
Figure 1 – binomial tree

$\text{ }$

Now consider a European option (either call or put) on the stock described above. When the stock goes up, we use $C_u$ to represent the value of the option. When the stock goes down, we use $C_d$ to represent the value of the option. The following is the binomial tree for the value of the option.

$\text{ }$
Figure 2 – option value tree

$\text{ }$

Replicating Portfolio
The key idea to price the option is to create a portfolio consisting of $\Delta$ shares of the stock and the amount $B$ in lending. At time 0, the value of this portfolio is $C=\Delta S + B$. At time $h$ (the end of the option period), the value of the portfolio is

$\text{ }$
Time $h$ value of the replicating portfolio

$\displaystyle \text{ } \left\{\begin{matrix} \displaystyle \Delta \times (dS \ e^{\delta h}) + B \ e^{r h}&\ \ \ \ \ \ \text{(when stock price goes down)}& \\ \text{ }&\text{ } \\ \Delta \times (uS \ e^{\delta h}) + B \ e^{r h}&\ \ \ \ \ \ \text{(when stock price goes up)} \end{matrix}\right.$

$\text{ }$

This portfolio is supposed to replicate the same payoff as the value of the option. By equating the portfolio payoff with the option payoff, we obtain the following linear equations.

$\text{ }$

$\displaystyle \text{ } \left\{\begin{matrix} \displaystyle \Delta \times (dS \ e^{\delta h}) + B \ e^{r h}=C_d&\ \ \ \ \ \ \text{ }& \\ \text{ }&\text{ } \\ \Delta \times (uS \ e^{\delta h}) + B \ e^{r h}=C_u&\ \ \ \ \ \ \text{ } \end{matrix}\right.$

$\text{ }$

There are two unknowns in the above two equations. All the other items – stock price $S$, dividend rate $\delta$, and risk-free interest rate $r$ – are known. Solving for the two unknowns $\Delta$ and $B$, we obtain:

$\text{ }$
$\displaystyle \Delta=e^{-\delta h} \ \frac{C_u-C_d}{S(u-d)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
$\text{ }$

$\displaystyle B=e^{-r h} \ \frac{u \ C_d-d \ C_u}{u-d} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$
$\text{ }$

Once the replication portfolio of $\Delta$ shares and $B$ in lending is determined, the price of the option (the value at time 0) is:

$\text{ }$
$C=\Delta S + B \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$
$\text{ }$

After plugging in (1) and (2) into (3), the option price formula becomes:

$\text{ }$
$\displaystyle C=\Delta S + B=e^{-r h} \biggl(C_u \ \frac{e^{(r-\delta) h}-d}{u-d} +C_d \ \frac{u-e^{(r-\delta) h}}{u-d} \biggr) \ \ \ \ \ \ \ \ \ (4)$
$\text{ }$

The price of the option described above is $C$, either given by formula (3) or formula (4). One advantage of formula (4) is that it gives the direct calculation of the option price without knowing $\Delta$ and $B$. Of course, if the goal is to create a synthetic option for the purpose of hedging or risk management, it will be necessary to know the make up of the replicating portfolio.

The $\Delta$ calculated in (1) is also called the hedge ratio and is examined in greater details in in this subsequent post.

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Examples

Example 1
Let’s walk through a quick example to demonstrate how to apply the above formulas. Suppose that the future prices for a stock are modeled with a one-period binomial tree with $u=$ 1.3 and $d=$ 0.8 and having a period of 6 months. The current price of the stock is $50. The stock pays no dividends. The annual risk-free interest rate is $r=$ 4%. • Determine the price of a European 55-strike call option on this stock that will expire in 6 months. • Determine the price of a European 45-strike put option on this stock that will expire in 6 months. The two-state stock prices are$65 and $40. The two-state call option values at expiration are$10 and $0. Apply (1) and (2) to obtain the replicating portfolio and then the price of the call option. $\text{ }$ $\displaystyle \Delta=\frac{10-0}{65-40}=\frac{10}{25}=$ 0.4 $\displaystyle B=e^{-0.04(0.5)} \ \frac{1.3(0)-0.8(10)}{1.3-0.8}=-16 e^{-0.02}=$ -$15.68317877

The replicating portfolio consists of holding 0.4 shares and borrowing $15.68317877. Call option price = $50 \Delta+B=$$4.316821227

$\text{ }$

The 2-state put option values at expiration are $0 and$5. Now apply (1) and (2) and obtain:

$\text{ }$
$\displaystyle \Delta=\frac{0-5}{65-40}=\frac{-5}{25}=-0.2$

$\displaystyle B=e^{-0.04(0.5)} \ \frac{1.3(5)-0.8(0)}{1.3-0.8}=13 e^{-0.02}=$ $12.74258275 The replicating portfolio consists of shorting 0.2 shares and lending$12.74258275.

Put option price = $50 \Delta+B=$ $2.742582753 $\text{ }$ Example 1 is examined in greater details in this subsequent post. More Examples Two more examples are in these previous posts: ___________________________________________________________________________________ What to do if options are mispriced What if the observed price of an option is not the same as the theoretical price? In other words, what if the price of a European option is not given by the above formulas? Because we can always hold stock and lend to replicate the payoff of an option, we can participate in arbitrage when an option is mispriced by buying low and selling high. The idea is that if an option is underpriced, then we buy low (the underpriced option) and sell high (the corresponding synthetic option, i.e. the replicating portfolio). On the other hand, if an option is overpriced, then we buy low (the synthetic option) and sell high (the overpriced option). Either case presents risk-free profit. We demonstrate with the options in Example 1. Example 2 • Suppose that the price of the call option in Example 1 is observed to be$4.00. Describe the arbitrage.
• Suppose that the price of the call option in Example 1 is observed to be $4.60. Describe the arbitrage. For the first scenario, we buy low (the option at$4.00) and sell the synthetic option at the theoretical price of $4.316821227. Let’s analyze the cash flows in the following table. $\text{ }$ Table 1 – Arbitrage opportunity when call option is underpriced $\left[\begin{array}{llll} \text{Expiration Cash Flows} & \text{ } & \text{Share Price = } \ 40 & \text{Share Price = } \ 65 \\ \text{ } & \text{ } \\ \text{Sell synthetic call} & \text{ } & \text{ } & \text{ } \\ \ \ \ \ \text{Short 0.4 shares} & \text{ } & - \ 16 & - \ 26 \\ \ \ \ \ \text{Lend } \ 15.683 & \text{ } & + \ 16 & + \ 16 \\ \text{ } & \text{ } \\ \text{Buy call } & \text{ } & \ \ \ 0 & \ \ \ 10 \\ \text{ } & \text{ } \\ \text{Total payoff} & \text{ } & \text{ } \ \ 0 & \ \ \ 0 \end{array}\right]$ $\text{ }$ The above table shows that the buy low sell high strategy produces no loss at expiration of the option regardless of the share prices at the end of the option period. But the payoff at time 0 is certain:$4.316821227 – $4.00 =$0.316821227.

For the second scenario, we still buy low and sell high. This time, buy low (the synthetic call option at $4.316821227) and sell high (the call option at the observed price of$4.60). Let’s analyze the cash flows in the following table.

$\text{ }$

Table 2 – Arbitrage opportunity when call option is overpriced

$\left[\begin{array}{llll} \text{Expiration Cash Flows} & \text{ } & \text{Share Price = } \ 40 & \text{Share Price = } \ 65 \\ \text{ } & \text{ } \\ \text{Buy synthetic call} & \text{ } & \text{ } & \text{ } \\ \ \ \ \ \text{Long 0.4 shares} & \text{ } & + \ 16 & + \ 26 \\ \ \ \ \ \text{Borrow } \ 15.683 & \text{ } & - \ 16 & - \ 16 \\ \text{ } & \text{ } \\ \text{Buy call } & \text{ } & \ \ \ 0 & - \ 10 \\ \text{ } & \text{ } \\ \text{Total payoff} & \text{ } & \text{ } \ \ 0 & \ \ \ 0 \end{array}\right]$

$\text{ }$

The above table shows that the buy low sell high strategy produces no loss at expiration of the option regardless of the share prices at the end of the option period. But the payoff at time 0 is certain: $4.60 –$4.316821227 = $0.283178773. These two examples show that if the option price is anything other than the theoretical price, there are arbitrage opportunities and there is risk-free profit to be made. ___________________________________________________________________________________ How to construct a binomial tree In the binomial tree in Figure 1, we assume that the share price at expiration is obtained by multiplying the original share price by the movement factors of $u$ and $d$. The binomial tree in Figure 1 may give the impression that the choice of the movement factors $u$ and $d$ is arbitrary as long as the up factor is greater than 1 and the down factor is below 1. In the next post, we show that $u$ and $d$ have to satisfy the following relation, else there will be arbitrage opportunities. $\displaystyle d < e^{(r-\delta) h} < u \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)$ Thus the choice of $u$ and $d$ cannot be entirely arbitrary. In particular the relation (5) shows that the future stock prices have to revolve around the forward price. $\displaystyle dS < Se^{(r-\delta) h} < uS \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)$ The purpose pf the factors $u$ and $d$ in the binomial tree is to incorporate uncertainty of the stock prices. In light of (6), we can set $u$ and $d$ by applying some volatility adjustment to $e^{(r-\delta) h}$. We can use the following choice of $u$ and $d$ to model the stock price evolution. $\displaystyle u = e^{(r-\delta) h \ + \ \sigma \sqrt{h}}$ $\displaystyle d = e^{(r-\delta) h \ - \ \sigma \sqrt{h}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7)$ where $\sigma$ is the annualized standard deviation of the continuously compounded stock return, $\sigma \sqrt{h}$ is the standard deviation of the continuously compounded stock return over a period of length $h$. The standard deviation $\sigma$ measures how certain we are that the stock return will be close to the expected return. There will be a greater chance of a return far from the expected return if the stock has a higher $\sigma$. If $\sigma=0$, then there is no uncertainty about the future stock prices. The formula (7) shows that when $\sigma=0$, the future stock price is precisely the forward price on the stock. When the binomial tree is constructed using (7), the tree will be called a forward tree. A note on calculation. If a problem does not specific $u$ and $d$ but assume a standard deviation of stock return $\sigma$, then assume that the binomial tree is the forward tree. We now use a quick example to demonstrate how to price an option using the forward tree. Example 3 Everything is the same as Example 1 except that the up and down stock prices are constructed using the volatility $\sigma=$ 30% (the standard deviation $\sigma$). The following calculates the stock prices at expiration of the option. $\displaystyle uS = 50 \ e^{(0.04-0) 0.5 \ + \ 0.3 \sqrt{0.5}}=$$63.06431255

$\displaystyle dS = 50 \ e^{(0.04-0) 0.5 \ - \ 0.3 \sqrt{0.5}}=$ $41.25989534 $\displaystyle u=\frac{63.06431255}{50}=$ 1.261286251 $\displaystyle d=\frac{41.25989534}{50}=$ 0.825197907 Using formulas (1), (2) and (3), the following shows the replicating portfolio and the call option price. Note that the binomial tree is based on a different assumption than that in Example 1. The option price is thus different than the one in Example 1. $\text{ }$ $\displaystyle \Delta=\frac{8.064312548-0}{63.06431255-41.25989534}=$ 0.369847654 $\displaystyle B=e^{-0.04(0.5)} \ \frac{1.261286251(0)-0.825197907(8.064312548)}{1.261286251-0.825197907}=$ –$14.95770971

The replicating portfolio consists of holding 0.369847654 shares and borrowing $14.95770971. Call option price = $50 \Delta+B=$$3.534672982

$\text{ }$

The following shows the calculation for the put option.

$\text{ }$
$\displaystyle \Delta=\frac{0-3.740104659}{63.06431255-41.25989534}=$ -0.171529678

$\displaystyle B=e^{-0.04(0.5)} \ \frac{1.261286251(3.740104659)-0.825197907(0)}{1.261286251-0.825197907}=$ $10.60320232 The replicating portfolio consists of shorting 0.171529678 shares and lending$10.60320232.

Put option price = $50 \Delta+B=$ $2.026718427 $\text{ }$ ___________________________________________________________________________________ More examples We present two more examples in illustrating the calculation in the one-period binomial option model where the stock prices are modeled by a forward tree. Example 4 The stock price follows a 6-month binomial tree with initial stock price$60 and $\sigma=$ 0.3. The stock is non-dividend paying. The annual risk free interest rate is $r=$ 4%. What is the price of a 6-month 55-strike call option? Determine the replicating portfolio that has the same payoff as this call option.

We will use risk-neutral probabilities to price the option.

$\displaystyle uS = 60 \ e^{(0.04-0) 0.5 \ + \ 0.3 \sqrt{0.5}}=$ $75.67717506 $\displaystyle dS = 60 \ e^{(0.04-0) 0.5 \ - \ 0.3 \sqrt{0.5}}=$$49.51187441

$\displaystyle C_u=$ 75.67717506 – 55 = 20.67717506

$\displaystyle C_d=$ 0

$\displaystyle u=\frac{75.67717506}{60}=$ 1.261286251

$\displaystyle d=\frac{49.51187441}{60}=$ 0.825197907

$\displaystyle p^*=\frac{e^{(0.04-0) 0.5} - 0.825197907}{1.261286251 - 0.825197907}=$ 0.447164974

$\displaystyle 1-p^*=$ 0.552835026

$\displaystyle C=(p^* \times C_u + (1-p^*) \times C_d) e^{-0.02}=$ 9.063023234

$\text{ }$

$\displaystyle \Delta=\frac{20.67717506-0}{75.67717506-49.51187441}=$ 0.790251766

$\displaystyle B=e^{-0.04(0.5)} \ \frac{1.261286251(0)-0.825197907(20.67717506)}{1.261286251-0.825197907}=$ –$38.35208275 The replicating portfolio consists of holding 0.79025 shares and borrowing$38.352.

$\text{ }$

Example 5
The stock price follows a 3-month binomial tree with initial stock price $40 and $\sigma=$ 0.3. The stock is non-dividend paying. The annual risk free interest rate is $r=$ 5%. What is the price of a 3-month 45-strike put option on this stock? Determine the replicating portfolio that has the same payoff as this put option. The calculation is calculated as in Example 3. $\displaystyle uS = 40 \ e^{(0.05-0) 0.25 \ + \ 0.3 \sqrt{0.25}}=$$47.05793274

$\displaystyle dS = 40 \ e^{(0.05-0) 0.25 \ - \ 0.3 \sqrt{0.25}}=$ $34.861374 $\displaystyle C_u=$ 0 $\displaystyle C_d=$ 45 – 34.861374 =$10.138626

$\displaystyle u=\frac{47.05793274}{40}=$ 1.176448318

$\displaystyle d=\frac{34.861374}{40}=$ 0.87153435

$\displaystyle p^*=\frac{e^{(0.05-0) 0.25} - 0.87153435}{1.176448318 - 0.87153435}=$ 0.462570155

$\displaystyle 1-p^*=$ 0.537429845

$\displaystyle C=(p^* \times C_u + (1-p^*) \times C_d) e^{-0.0125}=$ 5.381114117

$\text{ }$
$\displaystyle \Delta=\frac{0-10.138626}{47.05793274-34.861374}=$ -0.831269395

$\displaystyle B=e^{-0.05(0.25)} \ \frac{1.176448318(10.138626)-0.87153435(0)}{1.176448318 - 0.87153435}=$ $38.63188995 The replicating portfolio consists of shorting 0.831269395 shares and lending$38.63188995.

$\text{ }$

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Remarks

The discussion in this post is only the beginning of the binomial pricing model. The concepts and the formulas for the one-period binomial option model are very important. The one-period model may seem overly simplistic (or even unrealistic). One way to make it more realistic is to break up the one-period into multiple smaller periods and thus produce a more accurate option price. The calculation for the multi-period binomial model is still based on the calculation for the one-period model. Before moving to the multi-period model, we discuss the one-period model in greater details to gain more understanding of the one-period model.

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Practice problems

Practice Problems
Practice problems can be found in the companion problem blog via the following links:

basic problem set 1

basic problem set 2

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$\copyright \ \ 2015 \ \text{Dan Ma}$

## Put-Call Parity, Part 2

Put-call parity is a key idea in option pricing theory. It provides a tool for constructing equivalent positions. The previous post gives a general discussion of the put-call parity. In this post, we discuss the put-call parity for various underlying assets, i.e. the parity relations in this post are asset specific. The following is one form of the general put-call parity. This is the version (0) discussed in the previous post.

$\text{ }$
Put-Call Parity
$\displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$
$\text{ }$

The put-call parity has four components – the price of the call, the price of the put, the present value of the strike price and the present value of the forward price. In the general form of the put-call parity, the present value of the forward price completely take the dividends and time value of money into account. For a specific type of underlying asset, in order to make the put-call parity more informative, we may have to take all the interim payments such as dividends into account. Thus in the parity relations that follow, the general forward price is replaced with the specific forward price for that asset. Synthetic assets can then be created from the asset-specific put-call parity that is obtained.

The notations used here are the same as in the previous posts. The notation $F_{0,T}$ is the forward price. All contracts – forward and options and other type of contracts – are set at time 0 (today) and are to end at time $T$. The strike price for the options is $K$. The letter $r$ denotes the risk-free annual continuous interest rate. If the strike price $K$ is paid for an asset at time $T$, its present value at time 0 is $PV(K)=e^{-r T} K$. All options discussed here are European options, i.e. they can be exercised only at expiration.

All the parity relations that follow will obviously involve a call and a put. To make this extra clear, the call and the put in these relations have the same strike price and the same time to expiration. Thus whenever we say buying a call and selling a put, we mean that they are compatible in this sense.

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Put-call parity for stocks

Forward prices for stocks are discussed here. For a non-dividend paying stock, the forward price is $F_{0,T}=S_0 e^{r T}$, i.e. the price to pay for the stock in the future is the future value of the time 0 stock price. The following is the put-call parity of a non-dividend paying stock.

$\text{ }$
Put-Call Parity – non-dividend paying stock
$\displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S1)$
$\text{ }$

The parity (S1) says that there are two ways to buy a non-dividend paying stock at time 0. One is the outright stock purchase (the left side). The other way (the right hand side) is to buy a call, sell a put and lend the present value of the strike price $K$. By buying a call and selling a put, it is certain that you will buy the stock by paying $K$, which is financed by the lending of $PV(K)=e^{-r T} K$ at time 0. In both ways, you own the stock at time $T$. There is a crucial difference. In the outright stock purchase, you own the stock at time 0. In the “options” way, the stock ownership is deferred until time $T$. For the non-dividend paying stock, an investor is probably indifferent to the deferred ownership in the right hand side of (S1). For dividend paying stock, deferred ownership should be accounted for the parity equation.

$\text{ }$
Put-Call Parity – dividend paying stock (discrete dividend)
$\displaystyle S_0-PV(\text{Div})=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S2)$
$\text{ }$

In (S2), $\text{Div}$ refers to the dividends paid during the period from time 0 to time $T$ and $PV(\text{Div})$ refers to the time 0 value of $\text{Div}$. The deferred stock ownership on the right hand side of (S2) does not have the dividend payments while the outright stock ownership has the benefit of the interim dividend payments. Thus the cost of deferred stock ownership must be reduced by the amount of the dividend payments. This is why the dividend payments are subtracted on the left hand side. The next parity relation is for a stock or stock index paying continuous dividend.

$\text{ }$
Put-Call Parity – dividend paying stock (continuous dividend)
$\displaystyle S_0 e^{-\delta T}=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S3)$
$\text{ }$

Continuous dividends are reinvested (as additional shares) where $\delta$ is the annual continuous compounded dividend rate. The forward price is $F_{0,T}=S_0 e^{(r-\delta) T}$. The present value of the forward price is $S_0 e^{-\delta T}$, which is the left hand side of (S3). The left side of (S3) is saying that $e^{-\delta T}$ shares at time 0 will accumulate to 1 share at time $T$. The right hand side is saying that buying a call, selling a put and lending out the present value of $K$ at time 0 will lead to ownership of 1 share at time $T$.

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Synthetic stocks and other synthetic assets

In this section, we consider synthetic assets that can be created from the parity relations on stocks. These synthetic assets are parity relations. The left side of each of these relations is an asset that exists naturally in the financial market place. The right hand side is the synthetic asset – a portfolio that is an alternative asset that has the same cost and payoff, thus a portfolio that mimics the natural asset. For example, a synthetic stock is a combination of put and call and a certain amount of lending that will replicate the same payoff as owning a share of stock. In the next section, we will resume the discussion of put-call parity on underlying assets.

Each of the parity relation in this section is derived from an appropriate stock put-call parity by solving for the desired asset. For a synthetic stock, we put the stock on the left hand side by itself.

$\text{ }$
Synthetic stock – non-dividend paying
$\displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Syn1)$
$\text{ }$
Synthetic stock – discrete dividend paying
$\displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K+PV(\text{Div}) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Syn2)$
$\text{ }$
Synthetic stock – continuous dividend paying
$\displaystyle S_0 =(C(K,T)-P(K,T)+e^{-r T} K) \ e^{\delta T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Syn3)$
$\text{ }$

Note that (Syn1) is identical to (S1) since there is no dividend. The portfolio on the right hand side is the synthetic stock. For example, for (Syn2), the strategy of buying a call, selling a put, and lending out the present values of the strike price and the interim dividends is an alternative way to own a discrete dividend paying stock. There is a crucial difference between outright stock ownership on the left hand side and the deferred stock ownership on the right hand side. The synthetic stock pays no dividends. Thus the outright stock ownership is worth more than the synthetic stock. In other words, the cost of outright stock ownership exceeds the synthetic cost. By how much? By the present value of the interim dividends. This is why the present value of the dividend payments is added to the right hand side of (Syn2) and (Syn3).

Now we consider synthetic T-bills (or synthetic risk-free asset).

$\text{ }$
Synthetic T-bill – based on non-dividend paying stock
$\displaystyle e^{-r T} K=S_0-C(K,T)+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (T1)$
$\text{ }$
Synthetic T-bill – based on discrete dividend paying stock
$\displaystyle e^{-r T} K+PV(\text{Div})=S_0-C(K,T)+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (T2)$
$\text{ }$
Synthetic T-bill – based on continuous dividend paying stock
$\displaystyle e^{-r T} K=S_0 e^{-\delta T}-C(K,T)+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (T3)$
$\text{ }$

In (T1), (T2) and (T3), the right hand side is the synthetic way of creating a T-bill. Let’s look at (T3).

Relation (T3). In order to hold a synthetic T-bill, you buy $e^{-\delta T}$ shares of stock, sell a call and buy a put at time 0. At time $T$, the $e^{-\delta T}$ shares become 1 share, which will be used to meet the demand of either the call option or put option. If the stock price is more than $K$, the call buyer will want to exercise the call and you as a seller of the call will have to sell 1 share at the strike price $K$. If the stock price is less than $K$ at time $T$, you as the put buyer will want to sell 1 share of stock at the strike price $K$. So in either case, you have the amount $K$ at time $T$, precisely the outcome if you buy a T-bill with maturity value $K$.

Next we consider synthetic call options.

$\text{ }$
Synthetic call – based on non-dividend paying stock
$\displaystyle C(K,T)=S_0-e^{-r T} K+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (C1)$
$\text{ }$
Synthetic call – based on discrete dividend paying stock
$\displaystyle C(K,T)=S_0-e^{-r T} K-PV(\text{Div})+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (C2)$
$\text{ }$
Synthetic call – based on continuous dividend paying stock
$\displaystyle C(K,T)=S_0 e^{-\delta T}-e^{-r T} K+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (C3)$
$\text{ }$

The right hand side of the above three equations are synthetic ways to buy a stock call option. They can be derived by solving for $C(K,T)$ in the put-call parity relation in respective stock. It also pays to think through the cash flows on both sides. The right hand side of each of (C1) through (C3) consists of a leveraged position (stock purchase plus borrowing) and a long put to insure the leveraged position. For example, in the right hand side of (C1), borrow $e^{-r T} K$ and buy one share of stock (the leveraged position). Then use a purchased put to insure this leveraged position.

Another way to look at synthetic call is that the right hand side consists of a protective put and borrowing. A protective put is the combination of a long asset and a long put. For example, the right hand side of (C1) consists of $S_0+P(K,T)$ (a protective put) and the borrowing of $e^{-r T} K$, the present value of $K$.

Here’s the synthetic put options.

$\text{ }$
Synthetic put – based on non-dividend paying stock
$\displaystyle P(K,T)=C(K,T)-S_0+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (P1)$
$\text{ }$
Synthetic put – based on discrete dividend paying stock
$\displaystyle P(K,T)=C(K,T)-S_0+e^{-r T} K+PV(\text{Div}) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (P2)$
$\text{ }$
Synthetic put – based on continuous dividend paying stock
$\displaystyle P(K,T)=C(K,T)-S_0 e^{-\delta T}+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (P3)$
$\text{ }$

The right hand side of each of (P1) through (P3) is a synthetic put, a portfolio that mimics the payoff of a put option. Note that the right hand side consists of a long call and a short stock position (this is a protective call) and the lending of the present value of $K$.

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Put-call parity for currencies

A previous post on forward prices shows that the currency forward price is $F_{0,T}=x_0 \ e^{(r-r_f) T}$ where $x_0$ is the exchange rate (units of domestic currency per unit of foreign currency, e.g. dollars per euro), $r$ is the domestic risk-free rate and $r_f$ is the foreign currency risk-free rate. The present value of $F_{0,T}$ is then $e^{-r T} \ F_{0,T}=x_0 \ e^{-r_f T}$, which is the number of units of the domestic currency (e.g. dollars) at time 0 in order to have one unit of foreign currency (e.g. euro) at time $T$. Substituting $e^{-r T} \ F_{0,T}=x_0 \ e^{-r_f T}$ into the parity relation of (0), we have:

$\text{ }$
Put-Call Parity – Currencies
$\displaystyle x_0 \ e^{-r_f T}=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (F1)$
$\text{ }$
$\displaystyle x_0 \ e^{-r_f T}-e^{-r T} K=C(K,T)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (F2)$
$\text{ }$

In (F1) and (F2), we assume that the call and the put are denominated in dollars, i.e. both the strike price $K$ and the put premium and call premium are denominated in dollars. For ease of discussion, let’s say the foreign currency is euro. The premium $C(K,T)$ discussed here is in dollars and grants the right to pay $K$ to get 1 euro. The premium $P(K,T)$ discussed here is in dollars and grants the right to pay 1 euro to get $K$. Thus the strike price $K$ is an exchange rate of USD per euro.

For example, let’s say $K=$ 0.80 USD/Euro at time 0. If at time $T$ the exchange rate is $x_T=$ 0.9 USD/Euro, the call buyer would want to exercise the option by paying 0.8 USD for 1 euro. If at time $T$ the exchange rate is $x_T=$ 0.7 USD/Euro, then the long put position would want to exercise the put by paying 1 euro to get 0.8 USD.

The relation (F1) indicates that the difference in the call and put premiums plus lending the present value of the strike price is the same as lending the present value of the amount in dollars (the domestic currency) that is required to buy 1 euro at time $T$.

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Put-call parity for bonds

For a zero-coupon bond, the forward price is simply the future value of the bond price. For a coupon paying bond, the future price has to reflect the value of the coupon payments. In the following parity relations, $B_0$ is the bond price at time 0. The amount $PV(\text{Coupons})$ is the present value of the coupon payments made during the life of the options.

$\text{ }$
Put-Call Parity – zero-coupon bond
$\displaystyle B_0=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (B1)$
$\text{ }$
$\displaystyle B_0-PV(\text{Coupons})=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (B2)$
$\text{ }$

Note that for the zero-coupon bond, the parity relation is similar to the one for non-dividend paying stock.

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Summary

The following is the list of all the asset specific put-call parity relations discussed in this post.

$\text{ }$
Forward/Futures
$\displaystyle e^{-r T} \ F_{0,T}=C(K,T)-P(K,T)+PV(K)$
$\text{ }$

Non-dividend paying stock
$\displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

Discrete dividend paying stock
$\displaystyle S_0-PV(\text{Div})=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

Continuous dividend paying stock
$\displaystyle S_0 e^{-\delta T}=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

Currency
$\displaystyle x_0 \ e^{-r_f T}=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

Bond
$\displaystyle B_0=C(K,T)-P(K,T)+e^{-r T} K$
$\text{ }$

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$\copyright \ \ 2015 \ \text{Dan Ma}$

## Put-Call Parity, Part 1

Put–call parity is a relationship between the price of a European call option and European put option with the same strike price and time to expiration. It is one of the most important relationships in option pricing. It provides a tool for constructing equivalent positions. This post is a general discussion of put-call parity. In the next post, we discuss put-call parity in greater details for various underlying assets – e.g. stocks, treasuries and currencies.

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Synthetic forward – buying a call and selling a put

Suppose you follow the strategy of buying a call and selling a put (at time 0) where both options have the same underlying asset, the same strike price $K$ and the same time $T$ to expiration. At time $T$, it is certain that you will buy the underlying asset by paying the strike price $K$. Too see this, if at expiration of the options, the asset price is more than $K$, then you, as a call buyer will want to exercise the call option and pay $K$ to buy the asset. If the asset price at expiration is less than $K$, then you as a call buyer will not want to exercise but the put buyer that bought from you will want to exercise the put option. As a result, you will also buy the asset by paying the strike price $K$. Thus by entering into a long call and a short put (on the same underlying asset, with the same strike and same time to expiration), you will end up buying the underlying asset at time $T$ at the strike price $K$. What is being described sounds very much like a forward contract – a contract in which you can lock in a price today to pay for an asset a time $T$ in the future. For this reason, the strategy of buying a call and selling a put is called a synthetic forward contract.

___________________________________________________________________________________

Put-call parity

The above discussion on synthetic forward suggests that there are two ways to buy an underlying asset (e.g. a stock) at time $T$ in the future. They are:

1. Enter into a forward contract to buy the underlying asset by paying the forward price $F_{0,T}$ at time $T$.
2. Buy a call and sell a put today (on the same underlying asset, with the same strike price $K$ and the same time $T$ to expiration).

The two different strategies generate the same payoff. Hence they must have the same cost. Otherwise there would be arbitrage opportunities. By the “no-arbitrage pricing” principle, the net cost of the two strategies must equal. The cost at time 0 of the “buy call sell put” strategy is $C(K,T)-P(K,T)$, plus the present value of the strike price $K$, where $C(K,T)$ and $P(K,T)$ represent the call option premium and put option premium, respectively. The cost at time $T$ of the forward contract strategy is the forward price $F_{0,T}$. Thus cost at time 0 of the forward contract strategy is the present value of $F_{0,T}$. We can now equate the costs of the two strategies.

$\text{ }$
Put-Call Parity
$\displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$
$\text{ }$

The notation $PV(\cdot)$ denotes the time 0 value of an amount at the time $T$. Equation (0) is one form of the put-call parity, which is a statement that buying a call and selling a put is equivalent to a synthetic forward contract. It also tells us that buying a call and selling a put plus lending the present value of the strike price is equivalent to buying the underlying asset.

Other versions can be derived by algebraically rearranging equation (0), some of which have interesting interpretations. The following is one of them.

$\text{ }$
Put-Call Parity
$\displaystyle C(K,T)-P(K,T)=PV(F_{0,T}-K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
$\text{ }$

The left hand side of (1) is the net option premium – the premium paid for the call less the premium received for the put. When this amount is not zero, it is in effect the premium of the synthetic forward contract (this amount is the initial cash outlay for the synthetic forward contract). This is one difference between a synthetic forward and an actual forward. Note that an actual forward contract has zero premium (the initial cash outlay is zero). Another difference is that the “forward price” of the synthetic forward is the strike price $K$ of the options and while the forward price of the actual forward is $F_{0,T}$.

Suppose that the strike price $K$ is chosen to be less than the actual forward price $F_{0,T}$. Then the holder of the synthetic forward contract can buy the asset at a price lower than the forward price. This is certainly a benefit. In order to get this benefit, the holder of the synthetic forward contract has to pay the net option premium, which is the result of the call being more expensive than the put. In this scenario, the net payment is a little higher at time 0. As a result, the payment at time $T$ is a little less.

Suppose that the strike price $K$ is chosen to be more than the actual forward price $F_{0,T}$. Then the holder of the synthetic forward position is obliged to pay for the underlying asset at a price higher than the forward. It then makes sense for the holder of the synthetic forward position to be compensated by receiving a payment initially. This would occur if the put is more expensive than the call. In this scenario, the net payment is a little less at time 0, leading to a larger payment at time $T$.

If the strike price is chosen to be the same as the forward price $F_{0,T}$, then equation (1) suggests that the synthetic forward mimic exactly the actual forward (both have zero premium). For this to happen, premiums for the put and the call must be equal.

The right hand side of (1) is the value of the discount resulted from paying the strike price instead of the forward price. This version of the put-call parity says that the discount is identical to the net option premium.

___________________________________________________________________________________

Protective put and covered call

The next two versions can be interpreted in terms of a protective put and a covered call. A protective put consists of a long asset position and a long put. It is the strategy of buying a put option to protect against the risk of falling prices of a long asset position. A covered call consists of a long asset position and a short call. The covered call uses the upside profit potential of the long asset to back up (or cover) the call option sold to the call buyer. First, the protective call version:

$\text{ }$
Put-Call Parity
$\displaystyle PV(F_{0,T})+P(K,T)=C(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$
$\text{ }$

The left hand side of (2) is the time 0 cash outlay of buying the underlying asset and buying a put. The right hand side of (2) is time 0 cash outlay of buying a call option (with the same strike and time to expiration as the put) and buying a zero-coupon bond costing $PV(K)$. Thus equation (2) tells us that buying the underlying asset and buying a put on that asset (i.e. a protective put) have the same cost and generate the same payoff as the buying a call option and buying a zero-coupon bond. Adding a bond lifts the payoff graph but does not change the profit graph. Thus buying the asset and buying a put has the same profit as buying a call. Because of Equation (2), buying the underlying asset and buying a put is called a synthetic long call option. This point is also discussed in this previous post. Here’s the version of the put-call parity involving covered call.

$\text{ }$
Put-Call Parity
$\displaystyle PV(F_{0,T})-C(K,T)=PV(K)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$
$\text{ }$

The left hand side of (3) is the time 0 cash outlay of buying the underlying asset and selling a call on that asset (i.e. a covered call). The right hand side of (3) is the time 0 cash outlay of buying a zero-coupon bond costing $PV(K)$ and selling a put. Thus a covered call has the same cost and same payoff as buying a bond and selling a put. Once again, adding a bond does not change the profit. Thus a covered call has the same profit as selling a put. For this reason, a buying the underlying asset and selling a call is called a synthetic short put option. This point is also discussed in this previous post.

___________________________________________________________________________________

Summary

As a summary, we gather the various versions of the put-call parity in one place along with their interpretations.

$\text{ }$
Versions of Put-Call Parity
$\text{ }$
$\displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$
Interpretation: Time 0 cost of a long asset = Time 0 cost of (Long Call + Short Put + Long Bond).

$\text{ }$

$\displaystyle C(K,T)-P(K,T)=PV(F_{0,T}-K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
Interpretation: Net option premium (call option premium that is paid out less put option premium received) = the value of the discount as a result of paying the strike price instead of the forward price.
$\text{ }$

$\displaystyle PV(F_{0,T})+P(K,T)=C(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$
Interpretation: Time 0 cost of (Long Asset + Long Put) = Time 0 cost of (Long Call + Long Bond).
The portfolio on the left (Long Asset + Long Put) is called a protective put.
Because of (2), a protective put is considered a synthetic long call option.
$\text{ }$

$\displaystyle PV(F_{0,T})-C(K,T)=PV(K)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$
Interpretation: Time 0 cost of (Long Asset + Short Call) = Time 0 cost of (Long Bond + Short Put).
The portfolio on the left (Long Asset + Short Call) is called a covered call.
Because of (3), a covered call is considered a synthetic short put option.
$\text{ }$

In each of the above versions of parity, the portfolio of investments on the left side is equivalent to the portfolio of investment on the right side. More specifically, each version equates the costs of obtaining the portfolios at time 0. The bond indicated in the interpretations is a zero-coupon bond. A long position on a bond means lending.

One comment about the four parity relations discussed here. We derive the first one, which is version (0) by comparing the cash flows of two equivalent investments. The other three versions are then derived by algebraically rearranging the first version. As a learning device, it is a good idea to think through the cash flows and payoff of versions (2) through (3) independently of version (0). Doing so is a great practice and will help solidify the understanding of put-call parity. Drawing payoff diagrams can make the comparison easier. It is also possible to just think through the cash flows of both sides of the equation. For example,

let’s look at version (2). On the right side, you lend $PV(K)$ and buy a call at time 0. Then at time $T$, you get $K$ back. If the price of the underlying asset at that time is more than $K$, then you exercise the call – using the $K$ that you receive to buy the asset. So on the right hand, side, the payoff is $S_T-K$ if asset price is more than $K$ and the payoff is $K$ if asset price is less than $K$ (you would not exercise the call in this case). On the left hand side, you lend $PV(F_{0,T})$ and buy a put at time 0. At time $T$, you get $F_{0,T}$ back and you use it to pay for the asset. So you own the asset at time $T$. If the asset price at time $T$ is less than $K$, you exercise by selling the asset you own and receive $K$. Thus the payoff on the left hand side is $S_T-K$ if asset price is more than $K$ (in this case you don’t exercise the put and instead you profit from holding the asset). The payoff is $K$ if the asset price at time $T$ is less than $K$ (this is the case where you exercise the put option). The comparison shows that both sides of (2) have the same payoff at time $T$. Then it must be the case that they also have the same cost at time 0. Otherwise, there would be an arbitrage opportunity by buying the side that is low and sell the other side.

The basic put-call parity relations discussed in this post can be used in a “cookbook” fashion to create synthetic assets. For example, version (0) indicates that buying a call, selling a put and lending the present value of the strike price $K$ has the same cost and payoff as buying a non-dividend paying stock. Thus version (0) is a basis for constructing a synthetic stock. In the next post, we discuss the put-call parity for different underlying assets.

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$\copyright \ \ 2015 \ \text{Dan Ma}$

## Basic insurance strategies – protective put and protective call

This post follows up on a previous post, which is an introductory discussion on options. In this post, we focus on the two basic strategies of using options as insurance – protective put and protective call. These two strategies are for investors or traders who want to buy insurance to protect profits that come from holding either a long or short position. For every insurance buyer, there must be an insurance seller. In the next post, we discuss covered call and covered put – basic strategies for investors or traders who want to sell insurance protection.

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Option strategies that are insurance protection

Suppose that an investor has been holding an asset that has gained in value. The long asset position held by this investor will suffer a loss if prices fall. A long put option (a purchased put) has positive payoff when prices are less than the strike price. Buying a put option will guarantee a minimum sale price (the strike price of the put option) should the investor wishes to sell the asset at a future date. Thus the risk management strategy of buying a put option to guard against the loss of a long position is called a protective put.

A protective call deals with an opposite situation. Suppose an investor or trader is holding a short position on an asset (e.g. the investor has short sold a stock). The short asset position held by this investor will suffer a loss if prices increase. A purchased call option has positive payoff when the asset prices are greater than the strike price. When the investor buys a call option with the same underlying asset, the strike price is in effect a minimum purchased price of the asset should there be a price increase, thus keeping the loss at a minimum. Thus the risk management strategy of buying a call option to guard against the loss of a short position is called a protective call.

Protective puts and protective calls are basic insurance strategies that can be used to protect profits from either holding a long asset position or a short position. Both of these strategies will minimize the loss in the event that the prices of the asset position move in the wrong direction. Of course, the insurance protection comes at a cost in the form of an option premium, which is a cash fee paid by the buyer to the seller at the time the option contract is made. In the remainder of the post, we examine the protective put and the protective call in greater details by examining the payoff and profit diagrams.

The put and call considered here are European options, i.e., they can be exercised only at expiration.

___________________________________________________________________________________

Protective put

The protective put consists of a long asset position (e.g. owning a stock) and a long put option on the same asset. Our goal is to examine the payoff and profit of this combined position. We can then make some observations based on the profit diagram. Figure 1 is the payoff of the long asset. Figure 2 is the payoff of the long put option. Figure 3 is the payoff of the combined position. Figure 4 is the profit of long asset + long put. The strike price in all the diagrams is $K$. Instead of using a numerical example to anchor the diagrams, we believe that the following diagrams of payoff and the profit are clear. In fact, getting bogged down in a numerical example may make it harder to see the general idea. Asking questions such as – what happens when the asset is less than $K$, etc – will make the diagrams clear. In fact, reading the diagrams is a good concept check. An even better practice is to draw the payoff and profit diagrams on paper.

$\text{ }$

Figure 1 – Long Asset Payoff

$\text{ }$

Figure 1 is the payoff of the long asset. The strike price $K$ has no effect on the payoff of the long asset (Figure 1). The payoff of an asset is simply the value of the asset at a point in time. Thus the payoff is simply the asset price at a target date. The higher the asset price, the higher the payoff.

$\text{ }$
Figure 2 – Long Put Payoff

$\text{ }$

Figure 2 is the payoff of the long put option. For the long put, the payoff is $K-S_T$ when it makes sense for the put option buyer to exercise. Thus the payoff is positive to the left of $K$. To the right of $K$, the put option expires worthless, thus the payoff is 0. The sum of Figure 1 and Figure 2 gives Figure 3.

$\text{ }$
Figure 3 – Long Asset Long Put Combined Payoff

$\text{ }$

Figure 3 is the payoff of Long Asset + Long Put. The payoff to the right of the strike price $K$ is flat, which is a sign of the insurance at work. The positive payoff of the long put neutralizes the effect of falling prices of the long asset, minimizing the loss from holding a long asset when the prices go south. This position of long asset + long put will enjoy the upside potential in the event that prices go up. Of course, such as good insurance product is not free. The next diagram will take cost into account. First, the following formula shows the payoff of long asset + long put.

$\text{ }$

$\displaystyle \text{payoff of long asset + long put}=\left\{\begin{matrix} \displaystyle K&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ S_T&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\text{ }$

$\text{ }$
Figure 4 – Long Asset Long Put Combined Profit

$\text{ }$

Figure 4 is the profit of long asset + long put. Recall that the profit of a position is the payoff less the cost of acquiring that position. What is the cost of acquiring a long asset and a long put? The cost of the long asset is $S_0 e^{r T}$, the future value of the price paid at time 0. The cost of the long put is $P e^{r T}$, where $P$ is the put option premium paid by the buyer to the seller at time 0. Thus the cost of the long asset + long put is $(S_0+P) e^{r T}$. As a result, the profit graph is Figure 4 is obtained by pressing down the payoff in Figure 3 by the amount of the cost. The cost of the position is likely to be more than the strike price $K$. This is why $K-\text{Cost}$ in Figure 4 is negative.

Without the insurance (Figure 1), the long asset position will suffer substantial loss in the event that the prices are low. With insurance (Figure 4), the potential loss for the long asset position is essentially the put option premium. But the long asset position still enjoys the upside profit potential (less the option premium).

Another observation that can be made about Figure 4 is that its shape is very similar to the profit of a long call option (compare Figure 4 above with the Figure 1 in this previous post). The profit diagram of long asset + long put does not merely resemble the profit of a long call (with the same strike price); it is identical.

Based on Figure 4, the investment strategy of long asset + long put mimics the profit of the long call position (with the same strike price). Thus the position of long asset + long call is called a synthetic call option.

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Protective call

The protective call consists of a short asset position (e.g. shorting a stock) and a long call option on the same asset. We now examine the payoff and profit of this combined position. Figure 5 is the payoff of the short asset. Figure 6 is the payoff of the long call option. Figure 7 is the payoff of short asset + long call. Figure 8 is the profit of short asset + long call. The strike price in all the diagrams is $K$. From Figure 8, we will see that the combined position is a synthetic put option.

$\text{ }$

Figure 5 – Short Asset Payoff

$\text{ }$

Figure 5 is the payoff of the short asset position. Holder of a short asset position is concerned about rising prices of the asset. The holder of the short borrows the asset in a short sales and sells the asset immediately for cash, which is then accumulated at the risk-free rate. The short position will have to buy the asset back in the spot market at a future date to repay the lender. If the spot price at expiration is greater than the original sale price, then the short position will lose money. In fact the potential loss is unlimited.

$\text{ }$
Figure 6 – Long Call Payoff

$\text{ }$

Figure 6 is the payoff of the long call position. When the spot price at expiration is less than the strike price $K$, the call option expires worthless. When the spot price at expiration is greater than the strike price $K$, the payoff is $S_T-K$.

$\text{ }$
Figure 7 – Short Asset + Long Call Payoff

$\text{ }$

Figure 7 is the payoff of the combined position of a short asset and a long call. The payoff of the combined position is flat to the right of the strike price. This is a sign of the insurance at work. The upside potential of the short call is limiting the loss of the short asset position. The positive payoff of the long call is $S_T-K$. The payoff of the short asset is $-S_T$ when price is greater than the strike price. Then the combined payoff is $-K$ when price is greater than the strike price. To further clarify, the following is the payoff of the combined position.

$\text{ }$

$\displaystyle \text{payoff of short asset + long put}=\left\{\begin{matrix} \displaystyle -S_T&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ -K&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\text{ }$

$\text{ }$
Figure 8 – Short Asset + Long Call Profit

$\text{ }$

Figure 8 is the profit of short asset + long call. To derive the profit, we need to subtract the cost of acquiring the combined position from the payoff. The profit graph is Figure 8 is the result of lifting up the payoff graph. That suggests that in this case the profit is the payoff plus a positive amount. This is indeed correct since the cost of acquiring the position is a negative number. Thus subtracting the cost from the payoff is in effect adding a positive number.

To see the above point, the cost of acquiring the initial position is a positive number if it is a cash outflow (you pay to buy an asset) and is a negative number if it is a cash inflow (you sell an asset). In a short position, you borrow the asset and sell it to get cash, which is $-S_0 e^{r T}$ in this calculation. There is also the purchase of a call. Thus the total cost is $-S_0 e^{r T}+P e^{r T}$, which is likely a negative amount. So subtracting this negative cost from the payoff has the effect of lifting up the payoff graph.

Without the insurance of a long call (Figure 5), the short asset position has unlimited loss. With insurance (Figure 8), the loss of the short asset position is minimized, essentially the call option premium. The short asset position still enjoys the profit potential should asset prices fall (less the option premium).

Compare the above Figure 8 with the Figure 8 in this previous post, we see that they have the same shape. This is not coincidence. Both positions have the same profit. Thus the combined position of a short asset and a long call option is called a synthetic long put option since both have the same profit diagrams.

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Synthetic put and call

Just a couple of more observations to make about synthetic put and synthetic call.

Note that Figure 3 (the payoff of long asset + long put) also resembles the payoff of a long call option, except that the level part of the payoff is not at the x-axis. So Figure 3 is the lifting up of the usual long call option payoff by a uniform amount. That uniform amount can be interpreted as the payoff of a zero-coupon bond. Thus we have the following relationship.

$\text{ }$
payoff of “long asset + long put” = payoff of “long call + zero-coupon bond”
$\text{ }$

Adding a bond lifts the payoff graph. However, adding a bond to a position does not change the profit. To see this, simply subtract the cost of acquiring the position from the payoff. You will see that for the bond, the same amount appears in both the cost and the payoff. Thus we have:

$\text{ }$
profit of “long asset + long put” = profit of “long call”
$\text{ }$

As mentioned earlier, the above relationship indicates that the combined position of long asset + long put can be viewed as a synthetic long call. We now see that the protective put is identical to a long call.

Now similar thing is going on in a protective call. Note that Figure 7 resembles the payoff of a long put except that it is the pressing down of the payoff of a usual long put. We can think of this pressing down as a borrowing. Thus we have:

$\text{ }$
payoff of “short asset + long call” = payoff of “long put – zero-coupon bond”
$\text{ }$

Adding a bond means lending and subtracting a bond means borrowing. As mentioned before, adding or subtracting a bond lift or depress the payoff graph but does not change the profit graph. We have:

$\text{ }$
profit of “short asset + long call” = profit of “long put”
$\text{ }$

The above relationship is the basis for calling “short asset + long call” as a synthetic long put.

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$\copyright \ \ 2015 \ \text{Dan Ma}$